Unified Bulk-boundary Correspondence for Band Insulators*

The bulk-boundary correspondence provides for one of the most quintessential ideas in the topological classification of matter. However, the bulk-boundary correspondence has never been proven generally and has in certain scenarios been shown to fail to hold, depending on boundary profiles. In this regard we develop bulk numbers describing the exact number of in-gap modes without any of such subtleties in 1D. Based on this, we find a new 2D winding number, the pole winding number, that specifies the amount of robust metallic surface bands in the gap and their topological character. The underlying general methodology relies on a simple continuous extrapolation from the bulk to the boundary, while tracking the evolution of Green's function's poles in the vicinity of the bulk band edges. All the obtained numbers can be applied to the known insulating phases in a unified manner regardless of the specific symmetries. We directly apply our construction to various systems, including counterexamples for the conventional bulk-boundary correspondence in 1D, and predict the existence of boundary modes on various kinds of experimentally studied graphene edges. Finally, we sketch the 3D generalization of the pole winding number.